An identity for bipartite matching and symmetric determinant Original Research Article

Let A(x) = (Aij(x)) be an n × n symmetric polynomial matrix in x, and define δ(A) = degx det A(x), [the degree of the determinant of A(x)]. Let G(A) be the associated bipartite graph; the vertices of G(A) are identified with the rows and the columns of A, and the edges with the nonzero entries of A. We attach degx Aij(x) to edge (i,j) of G(A) as a weight and define image to be the maximum weight of a perfect matching in G(A). Then image, and the equality holds in the generic case, i.e., when the leading coefficients of Aij(x) (i ≤ j) are independent parameters. It is proven by a combinatorial argument that the gap between δ(A) and image, if any, can be resolved by a unimodular congruence transformation. That is, for a nonsingular A(x) there exists a unimodular U(x) such that A′(x) = U(x)A(x)U(x)T satisfies image. Note that if A(x) is transformed to A′(x) = U(x)A(x)U(x)T, δ(A) remains invariant, whereas image does change, since G(A) changes. The proof relies on the dual integrality of bipartite matching polytopes well known in polyhedral combinatorics.